Description: An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by AV, 3-Nov-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | cplgr0v.v | ⊢ 𝑉 = ( Vtx ‘ 𝐺 ) | |
cplgr2v.e | ⊢ 𝐸 = ( Edg ‘ 𝐺 ) | ||
Assertion | cplgr2v | ⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝐺 ∈ ComplGraph ↔ 𝑉 ∈ 𝐸 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cplgr0v.v | ⊢ 𝑉 = ( Vtx ‘ 𝐺 ) | |
2 | cplgr2v.e | ⊢ 𝐸 = ( Edg ‘ 𝐺 ) | |
3 | 1 | iscplgr | ⊢ ( 𝐺 ∈ UHGraph → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) ) |
4 | 3 | adantr | ⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝐺 ∈ ComplGraph ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) ) |
5 | 1 2 | uvtx2vtx1edgb | ⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝑉 ∈ 𝐸 ↔ ∀ 𝑣 ∈ 𝑉 𝑣 ∈ ( UnivVtx ‘ 𝐺 ) ) ) |
6 | 4 5 | bitr4d | ⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝐺 ∈ ComplGraph ↔ 𝑉 ∈ 𝐸 ) ) |